GCSE Maths Number Negative Numbers

Multiplying & Dividing Negative Numbers

Multiplying And Dividing Negative Numbers

Here we will learn about multiplying and dividing negative numbers including what negative numbers are and how to multiply and divide them.

There are also negative number worksheets and exam questions at worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are negative numbers?

Negative numbers are any numbers less than zero and have a negative sign (−) in front of them. 

Numbers greater than zero are referred to as positive numbers. If there is no sign in front of a number the number is positive.

On the number line below we can see some positive and negative integers (whole numbers):

The numbers in orange are negative and the blue numbers are positive.

Just like you can multiply and divide positive numbers, you can do the same with negative numbers. 

In order to multiply or divide negative numbers you must remember:

If the signs are the same, the answer is positive.

If the signs are different, the answer is negative.

When multiplying negative numbers:

The same rules apply for dividing negative numbers:

Click here to learn about adding and subtracting negative numbers.

What do you need to remember when multiplying and dividing negative numbers?

What do you need to remember when multiplying and dividing negative numbers?

How to multiply and divide negative numbers

In order to multiply and divide negative numbers:

  1. Multiply or divide the numbers normally.
  2. Change the sign as necessary using the rules of multiplying and dividing negative numbers:
    If the signs are the same, the answer is positive.
    If the signs are different, the answer is negative.

Explain how to multiply and divide negative numbers in 2 steps

Explain how to multiply and divide negative numbers in 2 steps

Multiplying negative numbers worksheet

Multiplying negative numbers worksheet

Multiplying negative numbers worksheet

Get your free worksheet of 20+ questions and answers. Includes reasoning and applied questions on multiplying and dividing negative numbers.

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Multiplying negative numbers worksheet

Multiplying negative numbers worksheet

Multiplying negative numbers worksheet

Get your free worksheet of 20+ questions and answers. Includes reasoning and applied questions on multiplying and dividing negative numbers.

DOWNLOAD FREE

Multiplying and dividing negative numbers examples

Example 1: multiplication of negative numbers

Multiply:

\[ -3\times5\]
  1. Multiply or divide numbers normally.

\[ 3\times5 = 15\]

2Change the sign using the rules of multiplying and dividing negative numbers:
If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.

\[-3\times5\]

In this case we have a number that is positive multiplied by a negative number (minus times positive gives a minus).

The signs are different so we must have a negative answer:

\[= -15\]

Example 2: division of negative numbers

Divide: 

\[ -24\div-6 \]

Multiply or divide numbers normally.

Change the sign using the rules of multiplying and dividing negative numbers:
If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.

Example 3: order of operations

Solve:

\[ -12\div(-6) +4\times(-2)\]

Multiply or divide numbers normally.

Change the sign using the rules of multiplying and dividing negative numbers:
If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.

Example 4: powers

Solve:

\[ (-2)^3\]

Multiply or divide numbers normally.

Change the sign using the rules of multiplying and dividing negative numbers:
If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.

Example 5: worded problem

The table below shows the temperatures recorded in Manchester at different times of the day. What is the product of the highest and lowest temperatures?

Time of dayTemperature (℃)
2am−6
7am−4
1pm2
6pm−2

Multiply or divide numbers normally.

Change the sign using the rules of multiplying and dividing negative numbers:
If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.

Common misconceptions

  • Greater negative doesn’t mean a larger number
    A common error is to assume that the larger a negative number is, the bigger number is.
    E.g.
    −3 is smaller than 2

  • Raising a negative number to a power greater than one
    Remember when raising a negative number to a power greater than 1 the resulting answer could be positive or negative.
    When you raise a negative number to an odd exponent the resulting answer is negative.
    When you raise a negative number to an even exponent the resulting answer is positive. 

Multiplying and dividing negative numbers is part of our series of lessons to support revision on negative numbers. You may find it helpful to start with the main negative numbers lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Practice multiplying and dividing negative numbers questions

1. Work out -12\times (-6)

-72
GCSE Quiz False

72
GCSE Quiz True

-18
GCSE Quiz False

18
GCSE Quiz False
12 \times 6=72

The signs are the same therefore the answer is positive:
-12\times (-6)=72

2. Work out 60 \div (-12)

-5
GCSE Quiz True

5
GCSE Quiz False

48
GCSE Quiz False

-48
GCSE Quiz False
60 \div 12 = 5

The signs are different therefore the answer is negative:
60 \div (-12)=-5

3. Work out -4 \div 2-7 \times (-8)

72
GCSE Quiz False

-58
GCSE Quiz False

54
GCSE Quiz True

-40
GCSE Quiz False

We must remember to apply BIDMAS here.
4 \div 2=2
The signs are different so it is -2

7 \times 8=56
We need to be really careful with the signs here. We are subtracting
7 \times (-8)
The signs are different so it is -56 .

The calculation becomes -2 –  – 56 .
There is – – together so we change this to a +

\begin{aligned} -2 – – 56 &= – 2 + 56\\ &=54 \end{aligned}

4. Work out: (-3)^{4}

12
GCSE Quiz False

81
GCSE Quiz True

-81
GCSE Quiz False

-12
GCSE Quiz False

(-3)^{4} means (-3) \times (-3) \times (-3) \times (-3)

Starting from the left,
(-3) \times (-3) = 9 (positive since the signs are the same)
9 \times (-3) = -27 (negative since the signs are different)
(-27) \times (-3) = 81 (positive since the signs are the same)

5. The temperature in Greenland was -8^{\circ}\text{C} . The temperature in Antarctica was 3 times as cold as it was in Greenland. What was the temperature in Antarctica?

-32^{\circ}\text{C}
GCSE Quiz False

-11^{\circ}\text{C}
GCSE Quiz False

24^{\circ}\text{C}
GCSE Quiz False

-24^{\circ}\text{C}
GCSE Quiz True

The calculation we need to do is 3 \times -8
3 \times 8 = 24

The signs are different so the answer is negative.
3 \times -8=-24

Multiplying and dividing negative numbers GCSE questions

The table shows the temperature in different cities across Canada.

City Temperature
Toronto −3
Calgary −12
Ottawa 2
Quebec City −6

(a) Which city has the lowest temperature?

(b) Find the product between the warmest and the coldest cities.

 

(3 marks)

Show answer

a)

Calgary

(1)

 

b)

Identifying the warmest and coldest temperatures (Ottawa and Calgary).

(1)

 

Correctly multiplying 2 and −12 to give −24

(1)

2. Mary has the following 6 cards:
 

She is going to choose 2 cards and multiply them.
 

(a) What is the largest possible number she can make?
 
(b) What is the smallest possible number she can make?

 

(4 marks)

Show answer

a)

For identifying 9 or 8.

(1)

 

Correctly multiplying 9 and 8 or 72 seen.

(1)

 

b)

For identifying −9 or 7.

(1)

 

Correctly multiplying two numbers or −63 seen.

(1)

3. The temperature in London was −6℃ on Wednesday. On the same day the temperature at the north pole was 4 times as cold as it was in London.
What was the temperature in the North Pole?

 

(2 marks)

Show answer

(−6) x 4 seen.

(1)

−24℃ (must have negative and celsius sign).

(1)

Learning checklist

You have now learned how to:

  • Multiply and divide integers both positive and negative
  • Use negative numbers in context and calculate intervals across 0

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